Subgroups — #27274795

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283357	C₂	1	1	yes	maximal, characteristic
2	#20273095	C₂	1	1	yes	maximal, characteristic
3	#20273095	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272551	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272551	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719945	C₂²	1	1	yes	characteristic
9	#16810682	—	2	2	no	—
10	#16810674	—	2	1	no	—
11	#16810674	—	2	1	no	—
12	#16810649	C₄	1	2	yes	—
13	#16810589	C₄	1	2	yes	—
14	C₄×#11719917	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810353	C₄	1	2	yes	—
18	#16810353	C₄	1	2	yes	—
19	#16810350	C₄	1	2	yes	—
20	#16810350	C₄	1	2	yes	—
21	#16810345	—	2	2	no	—
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#322	—	2	1	no	—
26	C₈²×C₅₃	—	2	1	no	—
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂²×#88322	—	2	1	no	—
30	C₂²×#88322	—	2	1	no	—
31	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
34	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
35	C₄×C₂²⋊D₅₃	—	2	1	no	—
36	C₄×C₂²⋊D₅₃	—	2	1	no	—
37	#13133176	C₂×C₄	1	1	yes	characteristic
38	#13133174	—	2	1	no	—
39	C₄²×D₅₃	—	2	1	no	—
40	#13133146	—	2	2	no	—
41	#13133146	—	2	2	no	—
42	#13133129	—	2	2	no	—
43	#13133129	—	2	2	no	—
44	#13133128	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	—	2	1	no	—
47	C₄×#88322	—	2	1	no	—
48	C₄×#88322	C₂×C₄	1	2	yes	—
49	C₄×#88322	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₂²⋊C₈×C₅₃	—	2	1	no	—
55	C₂²⋊C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₃	—	2	1	no	—
65	#11719945	—	2	4	no	—
66	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²⋊Dic₅₃	—	2	1	no	—
81	#11719921	—	2	1	no	—
82	#11719921	—	2	1	no	—
83	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
84	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
85	#11719919	—	2	1	no	—
86	#11719919	—	2	1	no	—
87	C₄×Dic₅₃	—	2	1	no	—
88	C₄×Dic₅₃	—	2	1	no	—
89	C₄×Dic₅₃	—	2	1	no	—
90	#11719917	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂³×C₅₃	#117	1	1	yes	characteristic
97	C₂²×D₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#131	1	4	yes	—
107	C₂²⋊D₅₃	—	4	2	no	—
108	C₂×Dic₅₃	—	2	1	no	—
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	—	2	1	no	—
111	D₂₁₂	—	2	2	no	—
112	C₄×D₅₃	—	4	2	no	—
113	C₄×D₅₃	—	4	2	no	—
114	C₄×D₅₃	—	4	2	no	—
115	Dic₁₀₆	—	2	2	no	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	—	2	2	no	—
118	C₈×C₅₃	—	4	2	no	—
119	C₈×C₅₃	—	4	2	no	—
120	#88322	—	2	4	no	—
121	#88322	—	4	2	no	—
122	#88322	—	4	2	no	—
123	#8189	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	#433	1	2	yes	—
128	C₂×D₅₃	—	4	1	no	—
129	C₂×D₅₃	—	4	1	no	—
130	C₂×D₅₃	—	4	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	#405	1	2	yes	—
134	C₄×C₅₃	#408	1	2	yes	—
135	C₄×C₅₃	#443	1	2	yes	—
136	C₄×C₅₃	#444	1	2	yes	—
137	Dic₅₃	—	4	1	no	—
138	Dic₅₃	—	4	1	no	—
139	Dic₅₃	—	4	1	no	—
140	#2922	—	53	1	no	—
141	#1476	—	53	1	no	—
142	#1476	—	53	1	no	—
143	#1450	D₅₃	1	1	yes	characteristic, complemented
144	C₂×#322	—	53	1	no	—
145	#1278	—	53	1	no	—
146	#1278	—	53	1	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	#1750	1	1	yes	characteristic
149	C₂×C₅₃	#1764	1	1	yes	characteristic
150	C₂×C₅₃	#1765	1	1	yes	characteristic
151	D₅₃	—	8	1	no	—
152	C₄×#41	—	53	1	no	—
153	#433	—	106	2	no	—
154	#431	—	106	1	no	—
155	#431	—	106	1	no	—
156	#431	C₂×D₅₃	1	1	yes	characteristic
157	#431	C₂×D₅₃	1	1	yes	characteristic
158	#424	—	53	2	no	—
159	#408	—	53	2	no	—
160	C₄×M₁₆	—	53	1	no	—
161	C₂×C₄×C₈	—	53	1	no	—
162	C₂×C₄×C₈	—	53	1	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	Q₈⋊C₈	—	53	2	no	—
165	Q₈⋊C₈	—	53	2	no	—
166	D₄⋊C₈	—	53	2	no	—
167	D₄⋊C₈	—	53	2	no	—
168	#322	—	2	1	no	—
169	#322	—	106	2	no	—
170	C₈²	—	2	1	no	—
171	C₅₃	#8189	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	53	1	no	—
173	C₂×M₁₆	—	53	2	no	—
174	C₂²×C₈	—	106	1	no	—
175	C₂²×C₈	—	106	1	no	—
176	C₂²×C₈	C₄×D₅₃	1	2	yes	—
177	C₄×Q₈	—	53	1	no	—
178	C₄×Q₈	—	53	1	no	—
179	C₄×D₄	—	53	1	no	—
180	C₄×D₄	—	53	1	no	—
181	C₄×D₄	—	106	1	no	—
182	C₄×D₄	—	106	1	no	—
183	#117	—	53	1	no	—
184	#117	—	106	1	no	—
185	C₂×C₄²	—	106	1	no	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	C₄⋊C₈	—	106	2	no	—
188	C₄⋊C₈	—	106	2	no	—
189	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
190	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	2	1	no	—
193	C₂²⋊C₈	—	106	2	no	—
194	C₂²⋊C₈	—	106	2	no	—
195	#97	—	106	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	53	2	no	—
199	C₄×C₈	—	53	2	no	—
200	C₄×C₈	—	106	1	no	—
201	C₄×C₈	—	106	1	no	—
202	C₄×C₈	—	106	1	no	—
203	C₄×C₈	D₂₁₂	1	2	yes	—
204	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
206	#41	—	106	4	no	—
207	C₂×Q₈	—	53	1	no	—
208	C₂×D₄	—	53	1	no	—
209	C₂×D₄	—	106	1	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	—	106	1	no	—
212	C₂²×C₄	—	106	1	no	—
213	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	#11719945	1	1	yes	characteristic
215	C₂²×C₄	#11719945	1	1	yes	characteristic
216	M₁₆	—	106	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	106	4	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	#11719921	1	2	yes	—
226	C₂×C₈	#11719921	1	2	yes	—
227	C₄⋊C₄	—	53	1	no	—
228	C₄⋊C₄	—	53	1	no	—
229	C₄⋊C₄	—	106	1	no	—
230	C₄⋊C₄	—	106	1	no	—
231	C₂²⋊C₄	—	106	1	no	—
232	C₂²⋊C₄	—	106	1	no	—
233	C₂²⋊C₄	—	106	1	no	—
234	C₄²	—	106	1	no	—
235	C₄²	—	106	1	no	—
236	C₄²	—	106	1	no	—
237	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂³	—	106	1	no	—
242	C₂³	#13133174	1	1	yes	characteristic
243	Q₈	—	106	2	no	—
244	D₄	—	106	2	no	—
245	D₄	—	212	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	106	1	no	—
249	C₂×C₄	—	106	1	no	—
250	C₂×C₄	—	106	1	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
256	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
260	C₂×C₄	#13133223	1	4	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	106	4	no	—
266	C₈	—	212	2	no	—
267	C₈	—	212	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	212	1	no	—
271	C₂²	—	212	1	no	—
272	C₂²	—	212	1	no	—
273	C₂²	C₄×#11719921	1	1	yes	characteristic
274	C₂²	#16810682	1	2	yes	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	212	1	no	—
278	C₄	—	212	1	no	—
279	C₄	—	212	1	no	—
280	C₄	#16810577	1	2	yes	—
281	C₄	#16810589	1	2	yes	—
282	C₄	#16810738	1	2	yes	—
283	C₄	#16810743	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	424	1	no	—
286	C₂	#20274482	1	1	yes	characteristic
287	C₂	#20274554	1	1	yes	characteristic
288	C₂	#20274561	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#8189
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₂×#16810345	C₂	semidirect
5. #20272551	C₂	semidirect
6. #20272551	C₂	semidirect
7. C₅₃×#1450	C₂	semidirect
143. #1450	D₅₃	semidirect
171. C₅₃	#8189	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20283357	2
#20273095	2
C₂×#16810345	2
#20272551	2
C₅₃×#1450	2
#8189	53